Minimum variance unbiased and moment estimators of carrier frequency offset in multi-carrier systems

ABSTRACT

A class of non data-aided cyclic based robust estimators for frequency offset estimation of multi-carrier systems is disclosed. The use of sufficient statistics provides a minimum variance unbiased (MVU) estimate of the frequency offset under complete knowledge of timing offset error. The Neyman-Fisher factorization theorem and Rao-Blackwell-Lehmann-Scheffe theorem are used to identify the sufficient statistic and appropriate mapping functions. It is shown that there is but one function of the sufficient statistics which results in the minimum variance estimate among the possible class of cyclic-based estimators. Also, a moment estimator of frequency offset is provided to obtain a consistent estimate of carrier offset under uncertain symbol timing error. The moment estimator does not rely on any probabilistic assumptions. Thus, its performance is insensitive to the distribution of the additive noise. A unified structure characterizing both the MVU and moment estimators, as well as a maximum likelihood estimator of a related, copending application is disclosed.

CROSS-REFERENCE TO RELATED APPLICATION

This application is related to a co-pending application entitled“Globally Optimum Maximum Likelihood Estimation of Joint CarrierFrequency Offset and Symbol Timing Error,” U.S. Ser. No. 09/496,890,filed on Mar. 2, 2000, assigned to the assignee of the instantinvention, and the disclosure therein is hereby incorporated byreference into the instant application.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for data communication ofsignals in units of a frame using an orthogonal frequency divisionmodulation (OFDM) algorithm, and in particular, to methods of estimatingcarrier frequency offset error at the receiver.

2. Description of the Prior Art

OFDM system is a viable modulation scheme for data transmission timevarying dynamic channels. However, it is known that performance of suchsystem is highly susceptible to non-ideal synchronization parameters.Specifically, symbol timing and carrier frequency offset become anincreasingly important issue in implementation of OFDM systems forpractical applications. It is known that carrier frequency offsetdeteriorates performance of OFDM systems by introducing interferenceamong the sub-channels. To overcome this imperfection, variouscompensation methods for estimation and correction of synchronizationparameters have been proposed. In order to compare the performance ofthese estimators, it is required to define a single number representingthe goodness of the estimate. Assuming that all estimators are unbiased,i.e., expectation of the estimate is equal to the parameter, thevariance of the estimator is used as a global measure for performancecomparison of these estimators.

Cramer-Rao lower bound (CRLB) is a fundamental lower bound on thevariance of the estimators and the unbiased estimator whose varianceequals CRLB is called efficient. When the evaluation of efficientestimator is not possible, it is desirable to obtain an estimator inwhich its performance becomes as close as possible to the CRLBfundamental bound. The estimator which is closest in performance to theCRLB estimator is known as a minimum variance unbiased (MVU) estimator.

Categorically, the previously proposed methods for synchronization ofOFDM systems can be classified into two main subclasses, namely minimummean square error (MMSE) and maximum likelihood (ML) estimators. In MMSEapproach, the estimator uses the information provided by the referencesignal (pilot tones) in order to minimize a cost function associatedwith the synchronization parameters. A salient feature of this approachis that no probabilistic assumptions are made with regard to the data.Although MMSE estimators usually result in a tractable (globally stable)and easy to implement realization, no optimal criteria (probabilistic)is associated with these estimators. Also, since part of the transmittedinformation is allocated to the reference pilots, the bandwidthefficiency of these methods is lower in comparison to the non-pilotschemes.

On the other hand, ML estimators provide the estimate of the unknownparameter subject to minimum probability of error criteria. Although notexactly efficient, ML estimators are asymptotically MVU, i.e., theirvariance attains that of MVU estimator as the length of data record goesto infinity. However, due to the physical constraints, systems withinfinitely long data records are not feasible for implementationpurposes.

P. H. Moose, in “A Technique for Orthogonal Frequency DivisionMultiplexing Frequency Offset Correction,” in IEEE Trans. OnCommunications, Vol. 42, No. 10, pp. 2908-2913, October 1994, describesthe use of a retransmission technique in order to reveal the frequencyoffset parameter in the likelihood function of the received signal. Dueto the redundancy introduced by repeating the data block, the data rateefficiency is decreased by a factor of two. To avoid this imperfection,a ML estimator based on cyclic prefix (CP) is described by J. van deBeck, M. Sandel and P. O. Borjesson, in “ML Estimation of Timing andFrequency Offset in OFDM Systems,” IEEE Trans. On Signal Processing,Vol. 45, No. 3, pp. 1800-1805, July 1997. In this approach, the sideinformation provided by the CP is used to obtain the likelihood functionfor joint estimation of symbol timing error and frequency offset in anOFDM system.

The likelihood function described in the Moose reference does notglobally characterize the observation vector over the entire range ofthe timing offset. Consequently, the ML estimator proposed based on thislikelihood function would result in considerable performance loss over afinite range of timing offset interval.

Currently, there is increasing interest in multi-carrier modulation(MCM) for dividing a communication channel into several subchannels andtransmitting many subcarriers through a single band using frequencydivision multiplexing (FDM) techniques. In the MCM method, however,because several subcarriers occupying a narrow frequency domain aretransmitted at one time, a relatively longer symbol period resultscompared with a single carrier modulation method. The MCM method has,owing to such characteristics, the advantages that equalization iseasily performed and that it has immunity to impulse noise. OFDM is atype of the MCM designed to maximize the working frequency efficiency bysecuring orthogonality among the multiplexed subcarriers. OFDM isapplied to mobile radio channels to attenuate multipath fading.

In an OFDM transmitting/receiving system, modulation and demodulation ofparallel data are carried out using the Fast Fourier Transform (FFT). Itis required that the sampled data be sent in predetermined frames,having passed through a FFT routine, been time-division multiplexed, andtransmitted, then restored at the receiving end. However, if thesynchronization is in error in the course of restoring the frame, thesignals demodulated after the FFT will be influenced by interchannel andintersymbol interference. Accordingly, the problem of synchronization inreforming the frame, especially any joint carrier frequency offset orsymbol timing error, must be addressed as a matter of importance.

Conventional synchronization methods as above-described encounterproblems in that the process of synchronization is not only verycomplex, but the synchronization is not realized rapidly.

SUMMARY OF THE INVENTION

Motivated by the sub-optimum performance of estimators of the prior art,a likelihood function for joint estimation of carrier frequency offsetand symbol timing error of OFDM systems is disclosed in the related,copending application; and a new optimum ML joint estimator is disclosedtherein. In order to reduce the variance of that ML estimator, a newclass of MVU estimators for frequency offset estimation of PFDM systemsis disclosed herein.

There is disclosed to exist but one function of sufficient statisticwhich provides the MVU estimate of the frequency offset. The estimatorprovided by the instant invention is a closed form expression; providingan estimator which is a function of data statistic. Consequently, itdoes not suffer from converging to multiple local minima, a problemwhich arises in ML technique with nonconvex loglikelihood functions.

The advantages of the instant MVU estimator over the class of previouslyproposed estimators are two; first, it is MVU, therefore its variance isminimum among the entire class of estimators which use the sameprobabilistic measure. Secondly, it provides a closed form expressionfor mapping the statistics into the estimation domain. The formerproperty assures optimality of the estimator, while the later facilitatethe closed loop analysis of the system.

Accordingly, the present invention is directed at a synchronizationmethod that substantially obviates on or more of the problems due tolimitations and disadvantages of the prior art. To achieve these andother advantages, and in accordance with the purpose of the invention asembodied and broadly described, there is provided a method of estimatingcarrier frequency offset error in a received sample bit stream includingan observation vector (OV), having an observed carrier frequency timingoffset c, and a plurality of data-symbol frames, having a symbol timingoffset error S. The method comprises the steps of generating aprobability density function (PDF) based on the OV and generating fromthe PDF an estimate of carrier frequency offset error, ε_(MVU|θ), beinga minimum variance unbiased (MVU) estimator.

In another aspect of the invention, the OV comprises an L-bit cyclicextension portion and a first and a second N-bit synchronization frame,and wherein the PDF comprises a first term, p1, based on the timingoffset

being within the span 1 to N and a second term, p2, based on said timingoffset

being within the span N+1 to N+L. In yet another aspect of the inventionthe received bit stream has uncorrelated independent identicallydistributed random signal and noise sequence variables with power ofσ_(s) ² and σ_(n) ², respectively, wherein the OV is denoted x, andwherein the MVU estimator ε_(MVU|) is the conditional expectation of asecond moment estimator, said second moment estimator given by$\overset{\Cup}{ɛ} = {\frac{\mathcal{J}}{2\pi}\ln\{ {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} \}}$

In another aspect of the invention the MVU estimator ε_(MVU|) is givenby $\begin{matrix}{{\overset{\Cup}{ɛ}}_{{MVU}❘\vartheta} = {E( {\overset{\Cup}{ɛ}❘_{T_{1}{({x,\vartheta})}}} )}} \\{= {\frac{\mathcal{J}}{2\pi}\ln\quad E\{ {{\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}}❘_{T_{1}{({x,\vartheta})}}} \}}} \\{= {\frac{1}{2\pi}\mathcal{J}\{ {\ln\frac{T_{1}( {x,\vartheta} )}{L\quad\sigma_{s}^{2}}} \}}}\end{matrix}$where ℑ is the imaginary operator and where${T_{1}( {x,\vartheta} )} = \{ \begin{matrix}{\sum\limits_{k = \vartheta}^{L + \vartheta - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} & {1 \leq \vartheta \leq N} \\{{\sum\limits_{k = 0}^{\vartheta - N - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} + {\sum\limits_{k = \vartheta}^{N + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} & {{N + 1} \leq \vartheta \leq {N + L}}\end{matrix} $

The invention also provides a method of synchronizing a received samplebit stream, comprising the steps of transmitting at a transmitter thebit stream including an observation vector (OV), receiving and samplingat a receiver the bit stream, the sampled bit stream including, the OVwith an observed carrier frequency offset ε, and a plurality ofdata-symbol frames, having a symbol timing offset error

; generating a probability density function (PDF) based on the OV;generating from the PDF an estimate of carrier frequency offset error,ε_(MVU|), being a minimum variance unbiased (MVU) estimator; andsynchronizing the received bit stream by the MVU estimate of carrierfrequency offset.

Further aspects of the synchronization method of the invention includewherein the OV comprises an L-bit cyclic extension portion and a firstand a second N-bit synchronization frame, and wherein the PDF comprisesa first term, p1, based on the observed timing offset

being within the span 1 to N and a second term, p2, based on theobserved timing offset

being within the span N+1 to N+L. In yet another aspect of theinvention, the synchronization method provides that the received bitstream has uncorrelated independent identically distributed randomsignal and noise sequence variables with power of σ_(s) ² and σ_(n) ²,respectively, wherein the OV is denoted x, and wherein the MVU estimatorε_(MVU|) is the conditional expectation of a second moment estimator,the second moment estimator given by$\overset{\Cup}{ɛ} = {\frac{\mathcal{J}}{2\pi}\ln\quad\{ {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} \}}$

Further aspect of the invention, the synchronization method wherein saidMVU estimator ε_(MVU|) is given by $\begin{matrix}{{\overset{\Cup}{ɛ}}_{{MVU}❘\vartheta} = {E( {\overset{\Cup}{ɛ}❘_{T_{1}{({x,\vartheta})}}} )}} \\{= {\frac{\mathcal{J}}{2\pi}\ln\quad E\{ {{\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}}❘_{T_{1}{({x,\vartheta})}}} \}}} \\{= {\frac{1}{2\pi}\mathcal{J}\{ {\ln\frac{T_{1}( {x,\vartheta} )}{L\quad\sigma_{s}^{2}}} \}}}\end{matrix}$where ℑ is the imaginary operator and where${T_{1}( {x,\vartheta} )} = \{ \begin{matrix}{\sum\limits_{k = \vartheta}^{L + \vartheta - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} & {1 \leq \vartheta \leq N} \\{{\sum\limits_{k = 0}^{\vartheta - N - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} + {\sum\limits_{k = \vartheta}^{N + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} & {{N + 1} \leq \vartheta \leq {N + L}}\end{matrix} $

In another aspect of the invention, A method of estimating carrierfrequency offset error is provided for a received sample bit streamincluding an observation vector (OV), having an observed carrierfrequency timing offset ε, and a plurality of data-symbol frames, the OVcomprising an L-bit cyclic extension portion and a first and a secondN-bit synchronization frame, comprising the steps of generating theexpected value of the autocorrelation of the kth entry of the OV; andgenerating from the expected value an estimate of carrier frequencyoffset error, ε_(mom), being a moment estimator.

In yet another aspect of the invention, the moment carrier frequencyoffset error estimation method further includes wherein the OV isdenoted x, and wherein the moment estimator ε_(mom) is given by${\hat{ɛ}}_{mom} = {\frac{\mathcal{J}}{2\pi}\{ {\ln\quad{T_{3}(x)}} }$where the statistic T₃(x) is defined as${T_{3}(x)} = {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = 0}^{N + L + 1}\quad{{x\lbrack k\rbrack}{{x^{*}\lbrack {k + N} \rbrack}.}}}}$

A method of synchronizing a received sample bit stream, is also providedby the invention comprising the steps of: transmitting at a transmittersaid bit stream including an observation vector (OV); receiving andsampling at a receiver said bit stream, said sampled bit streamincluding, said OV with an observed carrier frequency offset ε, and aplurality of data-symbol frames, having a symbol timing offset error

;

generating the expected value of the autocorrelation of the kth entry ofthe OV;

generating from the expected value an estimate of carrier frequencyoffset error, ε_(mom), being a moment estimator; and synchronizing thereceived bit stream by the moment estimate of carrier frequency offset.

Further aspect of the invention provides a synchronization methodwherein the OV is denoted x, and wherein the moment estimator ε_(mom) isgiven by${\hat{ɛ}}_{mom} = {\frac{\mathcal{J}}{2\pi}\{ {\ln\quad{T_{3}(x)}} }$where the statistic T₃(x) is defined as${T_{3}(x)} = {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = 0}^{N + L + 1}\quad{{x\lbrack k\rbrack}{{x^{*}\lbrack {k + N} \rbrack}.}}}}$

In yet another aspect of the invention, a method of estimating carrierfrequency offset error in a received sample bit stream including anobservation vector (OV), having an observed carrier frequency timingoffset ε, and a plurality of data-symbol frames, the OV comprising anL-bit cyclic extension portion and a first and a second N-bitsynchronization frame, comprising the steps of: selecting one estimatormethod from a group of three: a maximum likelihood (ML) estimator, aminimum variance unbiased (MVU) estimator, a moment estimator, based onsaid OV; and generating from the selected estimator an estimate ofcarrier frequency offset error.

Additional features and advantages of the invention will be set forth inthe description which follows, and in part will be apparent from thedescription, or may be learned by practice of the invention. Theobjectives and other advantages of the invention will be realized andattained by the methods particularly pointed out in the writtendescription and claims hereof, as well as the appended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a signal flow block diagram depicting the frequency recoveryloop for estimation of frequency offset.

FIG. 2 is a block diagram depiction of the ML estimator disclosed in thecross-referenced, copending application and the MVU and momentestimators of the instant invention.

FIG. 3 is a graph comparing the computer simulations of the prior art MLfrequency offset estimation and the ML frequency offset estimationdisclosed in the related, copending application.

FIG. 4 is a graph comparing the computer simulations of the MVUfrequency offset estimator of the instant invention with that of theCRLB estimator.

FIG. 5 graphically depicts the closed loop performance of the MVUestimator of the instant invention by comparing the frequency offsetestimate of the closed loop MVU estimator against an analyticalderivation thereof.

DETAILED DESCRIPTION OF THE INVENTION

The cross-referenced co-pending application entitled “Globally OptimumMaximum Likelihood Estimation of Joint Carrier Frequency Offset andSymbol Timing Error,” U.S. Ser. No. 09/496,890, filed on Mar. 2, 2000,incorporated herein by reference, discloses a probability densityfunction (PDF),p, which globally characterizes an observation vector xaccording to the equation $\begin{matrix}{{p( {x,ɛ,\vartheta} )} = {{{p_{1}( {x,ɛ,\vartheta} )}( {{U\lbrack {\vartheta - 1} \rbrack} - {U\lbrack {\vartheta - N - 1} \rbrack}} )} + {{p_{2}( {x,ɛ,\vartheta} )}( {{U\lbrack {\vartheta - N - 1} \rbrack} - {U\lbrack {\vartheta - N - L + 1} \rbrack}} )}}} & (14)\end{matrix}$where ε and

are the frequency offset and symbol timing error introduced by thesynchronization mismatch in the carrier frequency and symbol timing,respectively. And p₁ and p₂ are probability density functions derivedand described in the related copending application.

Further derived and described therein are the statistics T₁ and T₂$\begin{matrix}{{T_{1}( {x,\vartheta} )} = \{ {\begin{matrix}{\sum\limits_{k = \vartheta}^{L + \vartheta - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}} & {1 \leq \vartheta \leq N} \\{{\sum\limits_{k = 0}^{\vartheta - N - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}} +} & {{N + 1} \leq \vartheta \leq {N + L}} \\{\sum\limits_{k = \vartheta}^{N + L - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}} & \quad\end{matrix}{and}} } & (15) \\{{T_{2}( {x,\vartheta} )} = \{ \begin{matrix}{{\sum\limits_{k = \vartheta}^{L + \vartheta - 1}{{x\lbrack k\rbrack}}^{2}} + {{x\lbrack {k + N} \rbrack}}^{2}} & {1 \leq \vartheta \leq N} \\{{\sum\limits_{k = 0}^{\vartheta - N - 1}{{x\lbrack k\rbrack}}^{2}} + {{x\lbrack {k + N} \rbrack}}^{2} +} & {{N + 1} \leq \vartheta \leq {N + L}} \\{{\sum\limits_{k = \vartheta}^{N + L - 1}{{x\lbrack k\rbrack}}^{2}} + {{x\lbrack {k + N} \rbrack}}^{2}} & \quad\end{matrix} } & (16)\end{matrix}$N and L being the number of samples per frame and the number ofcyclically-extended samples appended at the beginning of each frame,respectively, all as further described in the related, copendingapplication.

The MVU estimator of the instant invention is derived by resorting tothe theory of sufficient statistics. The first step in deriving the MVUestimator is to obtain the sufficient statistic for the PDF given in(14). The sufficient statistic is known to be a function of theobservation vector, namely T(x), such that the conditional PDF of theobservation vector given T(x) does not depend on the unknown estimationparameters [ε,

]. Evaluating the sufficient statistic is a formidable task for thebroad class of PDFs, however the Neyman-Fisher Factorization theorem canbe used for identifying the potential sufficient statistics. Accordingto this theorem, if the PDF can be factored in the form g(T(x), ε,

,)h(x) where g is a function depending on x only through T(x) and h(x)is a function depending only on x, then T(x) is a sufficient statisticfor estimation of the parameters ε and

. By reformulating the PDF given in 14 to${p( {x,ɛ,\vartheta} )} = {{\mathbb{e}}^{\frac{{2{\mathcal{R}{\lbrack{{\mathbb{e}}^{j2\pi ɛ}{T_{1}{({x,\vartheta})}}{a}}\rbrack}}} - {{a}^{2}{T_{2}{({x,\vartheta})}}}}{2{({1 - {a}^{2}})}{({\sigma_{s}^{2} + \sigma_{n}^{2}})}}}{h_{1}(x)}}$there is a direct dependency between the parameter

and the statistics T₁(x,

) and T₂(x,

). Based on this observation, the Neyman-Fisher theorem fails to providea sufficient statistic for estimation of

, we can factor the PDF into $\begin{matrix}{{p( {x,{ɛ❘_{\vartheta}}} )} = {{\mathbb{e}}^{\frac{\mathcal{R}{\lbrack{{\mathbb{e}}^{j2\pi ɛ}{T_{1}{({x,\vartheta})}}{a}}\rbrack}}{{({1 - {a}^{2}})}{({\sigma_{s}^{2} + \sigma_{n}^{2}})}}}{h_{2}(x)}}} & (17)\end{matrix}$Clearly then, T₁(x,

) forms a sufficient statistic for estimation of the parameter ε.

Next, application of the Rao-Blackwell-Lehmann-Scheffe (RBLS) theorem isused to find the MVU estimator. According to this theorem, if {haeckover (ε)} is an unbiased estimator of ε and T(x) is a sufficientcomplete statistic for ε then {circumflex over (ε)}=Ε({haeck over(ε)}|_(T(x))) is a valid, unbiased, MVU estimator of ε.

In applying the above theorem, we need to obtain an unbiased estimatorof ε, termed {circumflex over (ε)}, and determine the conditionalexpectation of this estimator given the statistic T₁(x,

). An appropriate candidate for the unbiased estimator of ε can beobtained from the statistical moments of the random vector x. Theautocorrelation function of the observation vector x, Ε, yields that thesecond moment of the random variable x[k] with kεΩ, satisfies thefollowing identityΕ{x[k]x*[k+N]}=σ _(s) ² e ^(−j2πε)  (18)Having this observation, one uses the second moment estimator as anunbiased estimator for ε as given by $\begin{matrix}{\overset{\Cup}{ɛ} = {\frac{\mathfrak{J}}{2\pi}\ln\{ {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}}} \}}} & (19)\end{matrix}$where ℑ is the imaginary operator. In deriving the above estimator,E(x[k]x*[k+N]) was replaced by its natural estimator$\frac{1}{L}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}{{x\lbrack k\rbrack}x*{\lbrack {k + N} \rbrack.}}}$It is straightforward to verify that this estimator is unbiased as itsatisfies the condition $\begin{matrix}{{E\{ \overset{\Cup}{e} \}} = {{\frac{\mathfrak{J}}{2\pi}\ln\quad E\{ {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}}} \}} = ɛ}} & (20)\end{matrix}$

Next, one obtains the conditional expectation of {circumflex over (ε)}given the sufficient statistic T_(1(x,)) as follows $\begin{matrix}\begin{matrix}{{\overset{\Cup}{ɛ}}_{{MVU}❘\vartheta} = {E( {\overset{\Cup}{ɛ}❘_{T_{1}{({x,\vartheta})}}} )}} \\{= {\frac{\mathfrak{J}}{2\pi}\ln\quad E\{ {{\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}}}❘_{T_{1}{({x,\vartheta})}}} \}}} \\{= {\frac{1}{2\pi}{\mathfrak{J}}\{ {\ln\quad\frac{T_{1}( {x,\vartheta} )}{L\quad\sigma_{s}^{2}}} \}}}\end{matrix} & (21)\end{matrix}$

It is important to emphasize that since the underlying PDF given in (14)belongs to the exponential family of PDFs, then the sufficientstatistics T₁(x,

) forms a complete statistic for estimation of the parameter ε.Therefore, the mapping function obtained from applying RBLS theorem,namely lnT(x,

), is but one function of the statistic T₁(x,

) and no other estimator with the same statistic can result in a lowervariance with respect to MVU estimator.

A. Cramer-Rao Lower Bound

It is known that under broad conditions, the variance of any unbiasedestimator of a nonrandom parameter ε satisfies the CRLB as$\begin{matrix}{{{var}( {\overset{\Cup}{ɛ}}_{{MVU}❘\vartheta} )} \geq \frac{1}{I(ɛ)}} & (22)\end{matrix}$where I(ε) is the Fisher Information given by $\begin{matrix}{{I(ɛ)} = {- {E\lbrack \frac{{\partial^{2}\ln}\quad{{pr}( {x,{ɛ❘_{\vartheta}}} )}}{\partial ɛ^{2}} \rbrack}}} & (23)\end{matrix}$

Substituting (17) into (23), after some algebraic manipulations, theCRLB of the MVU estimator becomes $\begin{matrix}{{{var}( {\overset{\Cup}{ɛ}}_{{{MVU}}_{\vartheta}} )} = {\frac{( {1 - {a}^{2}} )( {\sigma_{s}^{2} + \sigma_{n}^{2}} )}{( {2\pi} )^{2}{a}E\{ {T_{1}( {x,\vartheta} )} \}} = \frac{( {1 + \frac{1}{SNR}} )^{2} - 1}{( {2\pi} )^{2}L}}} & (24)\end{matrix}$where ${SNR}\overset{\Delta}{=}\frac{\sigma_{S}^{2}}{\sigma_{N}^{2}}$is the signal to noise ratio at the receiving end.B. Closed Loop Performance

FIG. 1 is a signal flow block diagram depicting the frequency recoveryloop for estimation of frequency offset according to the instantinvention. A closed loop system is obtained by feeding back theinformation obtained from the estimator into a sampler block(bootstrap). A sampler 10 updates its frequency at the beginning of eachobservation vector (every (N+L) samples). The sampler 10 receives thelatest signal and produces the sampled symbol stream x[k] therefromwhich is conducted to a block 12 and a block 14 performing theautocorrelation operation on the observation vector as required byequation (19). The result of the autocorrelation operation is conductedto a moving average (MA) filter block 16 and therefrom to a block 18performing the natural logarithm function and thence to a block 20performing the imaginary operation function, as required by equation(19). To match the various sampling frequencies used in the system, adown sampler (decimator) block 22 is used prior to return to the sampler10 to produce the minimum variance unbiased estimator, {circumflex over(ε)}[m]. Finally, a gain block (G) 24 is used to control the closed loopcharacteristic of the system (stability, settling time, noisesensitivity).

According to FIG. 1, the frequency offset for the mth observation vectorcan be expressed as $\begin{matrix}\begin{matrix}{{\hat{ɛ}\lbrack m\rbrack} = {\frac{1}{2\pi}\ln\frac{1}{L\quad\sigma_{S}^{2}}{\sum\limits_{i = 0}^{L - 1}{{x_{({m - 1})}\lbrack i\rbrack}{\mathbb{e}}^{{- j}\frac{2\pi\quad{\hat{ɛ}{\lbrack{m - 1}\rbrack}}{({\vartheta + i})}}{N}G}}}}} \\{{x_{({m - 2})}^{*}\lbrack i\rbrack}{\mathbb{e}}^{j\frac{2\pi\quad{\overset{\Cap}{ɛ}{\lbrack{m - 2}\rbrack}}{({\vartheta + i})}}{N}G}} \\{= {\frac{1}{2\pi}\ln\quad\frac{1}{L\quad\sigma_{s}^{2}}e_{j\frac{2\pi\quad\hat{\Delta}\quad{ɛ{\lbrack{m - 1}\rbrack}}{(\vartheta)}}{N}}{\sum\limits_{i = 0}^{L - 1}{{x_{({m - 1})}\lbrack i\rbrack}{x_{({m - 2})}^{*}\lbrack i\rbrack}}}}} \\{{\mathbb{e}}^{{- j}\frac{2\pi\quad\Delta\quad{\overset{\Cap}{ɛ}{\lbrack{m - 1}\rbrack}}{(i)}}{N}}}\end{matrix} & (25)\end{matrix}$where Δ{circumflex over (ε)}[m−1]

(ε[m−1]−ε)[m−2])G and x_(m)[i]

x[m(N+L)+

+i]. The term inside the sum is a stochastic quantity and does not havea closed form expression. However, for reasonably high signal to noiseratio it can be well approximated by its expected value(Ε[x_((m−1))[i]x*_((m−2))[i]]=σ_(s) ²). Therefore, the expression insidethe sum can be written as $\begin{matrix}\begin{matrix}{{\sum\limits_{i = 0}^{L - 1}{\mathbb{e}}^{{- j}\frac{2\pi\quad\Delta\quad{ɛ{\lbrack{m - 1}\rbrack}}{(i)}}{N}}} = \frac{1 - {\mathbb{e}}^{{- j}\frac{2\pi\quad\Delta\quad{\overset{\Cap}{ɛ}{\lbrack{m - 1}\rbrack}}L}{N}}}{1 - {\mathbb{e}}^{{- j}\frac{2\pi\quad\Delta\quad{\overset{\Cap}{ɛ}{\lbrack{m - 1}\rbrack}}}{N}}}} \\{= {{\mathbb{e}}^{{- j}\frac{2\pi\quad\Delta\quad{ɛ{\lbrack{m - 1}\rbrack}}{({L - 1})}}{2N}}\frac{\sin( \frac{\Delta\quad{\hat{ɛ}\lbrack {m - 1} \rbrack}L}{2N} )}{\sin( \frac{\Delta\quad{\hat{ɛ}\lbrack {m - 1} \rbrack}}{2N} )}}} \\ {\Delta\quad{\hat{ɛ}\lbrack {m - 1} \rbrack}}arrow{0{~~~}{\mathbb{e}}^{{- j}\frac{2\pi\quad\Delta\quad{\hat{ɛ}{\lbrack{m - 1}\rbrack}}}{2N}L}} \end{matrix} & (26)\end{matrix}$Substituting (26) into (25), after some algebraic manipulations, thefrequency offset of m'th observation vector becomes $\begin{matrix}{{\hat{ɛ}\lbrack m\rbrack} = {{- ( {{\hat{ɛ}\lbrack {m - 1} \rbrack} - {\hat{ɛ}\lbrack {m - 2} \rbrack}} )}\frac{G}{N}( {\vartheta + \frac{L - 1}{2}} )}} & (27)\end{matrix}$The above equation represents a second order finite difference system inwhich its dynamic can be obtained from solving the following equation{circumflex over (ε)}[m]+β{circumflex over (ε)}[m−1]−β{circumflex over(ε)}[m−2]=0where$\beta\overset{\Delta}{=}{\frac{G}{N}{( {\vartheta + \frac{L - 1}{2}} ).}}$Clearly, the solution to the above finite difference equation is theform of{circumflex over (ε)}[m]=c ₁(z ₁)^(m) +c ₂(z ₂)^(m)   (28)where$z_{1,2}\overset{\Delta}{=}\frac{{- \beta} \pm \sqrt{\beta^{2 +}4\beta}}{2}$are two dynamical modes of the system. The smaller root (negative)results in a high frequency oscillation in the frequency offsetestimate. However, as is shown in a computer simulation, infra, thisterm is filtered out by the moving average filter. To assure stability,the gain block should be set such that both poles lie inside the unitcircle.|β_(max)+{square root}{square root over (β² max+4β max|)}≦2   (29)where $\begin{matrix}{\beta_{\max}\overset{\Delta}{=}{{\max\limits_{\vartheta}{\frac{G}{N}( {\vartheta + \frac{L - 1}{2}} )}} = {\frac{G}{N}( {N + L + \frac{L - 1}{2}} )}}} & (30)\end{matrix}$

It is known that when the timing offset parameter is not known to thereceiver or if the noise PDF differs from Gaussian distribution, findingthe optimum estimator (ML, MVU, CRLB) may not be an easy task. Howeverthere exists a moment estimator which provides a consistent estimate forestimation of frequency offset regardless of noise distribution andtiming offset values. Although there is no optimum criterion associatedwith the moment estimator due to its simple structure, it is frequentlyused as an initial estimate for other estimators such as ML estimator.Consider a sequence of first N+L samples of vector x. Using theautocorrelation of kth entry of this vector satisfies the followingidentity Moment Estimator $\begin{matrix}{{r_{xx}\lbrack N\rbrack} = \{ \begin{matrix}{\sigma_{s}^{2}{\mathbb{e}}^{{- j}\quad 2\pi\quad ɛ}} & {k \in \Omega} \\0 & {k \notin \Omega}\end{matrix} } & (31)\end{matrix}$Using Base Rule, the expected value of the above function (with respectto parameter k) can be expressed as $\begin{matrix}\begin{matrix}{{E_{k}\lbrack {r_{xx}\lbrack N\rbrack} \rbrack} = {{\sigma_{s}^{2}\exp^{- {j2\pi ɛ}}{{pr}( {k \in \Omega} )}} + {0\quad{{pr}( {k \notin \Omega} )}}}} \\{= {\frac{L}{L + N}\sigma_{s}^{2}\exp^{- {j2\pi ɛ}}}}\end{matrix} & (32)\end{matrix}$Substituting the Nth autocorrelation lag with its natural estimator, themoment estimator for frequency offset under uncertain timing offset canbe found as $\begin{matrix}{{\hat{ɛ}}_{mom} = {\frac{\mathfrak{J}}{2\pi}\{ {\ln\quad{T_{3}(x)}} \}}} & (33)\end{matrix}$where the statistic T₃(x) is defined as $\begin{matrix}{{T_{3}(x)} = {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = 0}^{N + L + 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}}}} & (34)\end{matrix}$Statistical assessment of moment estimator is a formidable task overentire range of SNR. However, for relatively high SNR, the randomobservation vector is heavily concentrated about its mean. Using thestatistical linearization, and a first-order Taylor expression of theestimator about its mean to obtain the variance of estimate. In doingso, one substitutes for the random variable x in (34) and obtains$\begin{matrix}\begin{matrix}{{T_{3}( \lbrack x\rbrack )} = {f( \lbrack {s,w} \rbrack )}} \\{= {\frac{\mathfrak{J}}{\ln}\{ {L\quad\sigma_{s}^{2}{\sum\limits_{k = 0}^{N + L - 1}( {{{s\lbrack {k - \vartheta} \rbrack}\exp^{j\frac{2\pi\quad k\quad ɛ}{N}}} + \quad{w\lbrack k\rbrack}} )}} }} \\ ( {s*\lbrack {k - \vartheta + \quad N} \rbrack\quad\exp^{{{- j}\frac{2{\pi{({k + N})}}ɛ}{N}} + {w*{\lbrack{k + N}\rbrack}}}} ) \}\end{matrix} & (35)\end{matrix}$where the signal (s) and noise (w) vector are defined ass

[s[0]] . . . s[N+L−1]]w

[w[0]] . . . w[N+L−1]]  (36,37)

By virtue of the above equation, the expected value of observationvector for a fixed realization of signal vector s would beΕ[ε_(mom)]=ƒ(s). Then performing a first order Taylor expansion ofƒ([s.w]) about the point Εw_([x]) yields $\begin{matrix}\begin{matrix}{ {{\hat{ɛ}}_{mom} = {{{f( \lbrack {s,0} \rbrack )} + {{\nabla{{wf}( \lbrack {s,0} \rbrack )}}*w}} = {f( {s,0} \rbrack}}} ) +} \\{ {\sum\limits_{n = 0}^{L + N - 1}\frac{\partial{f( \lbrack {s,w} \rbrack )}}{\partial{w\lbrack n\rbrack}}} \middle| ( {w = 0} )^{w{\lbrack n\rbrack}} }\end{matrix} & (38)\end{matrix}$Taking the derivative of (35) with respect to w[n] and setting w=0,results in $\begin{matrix}{{{\frac{\partial h}{\partial{w\lbrack n\rbrack}}}w} = {0 = {\frac{1}{2\pi}( {\sum\limits_{i = 0}^{N + L - 1}{{s\lbrack {i - \vartheta} \rbrack}s*\lbrack {i - \vartheta + N} \rbrack{\mathbb{e}}^{- {j2\pi ɛ}}}} )^{- 1}}}} \\{s*\lbrack {n + N - \vartheta} \rbrack{\mathbb{e}}^{\frac{2\pi\quad{ɛ{({n + N})}}}{N}}}\end{matrix}$

The second term in (38) represents the contribution of noise in theestimate. Knowing that noise samples are iid with power of σ_(w) ², thevariance of estimate can be obtained from $\begin{matrix}{{{var}(ɛ)} = \frac{\sum\limits_{n = 0}^{L + N - 1}{{{s\lbrack {n + N - \vartheta} \rbrack}}^{2}\sigma_{w}^{2}}}{( {2\pi{\sum\limits_{i = 0}^{N + L - 1}{{s\lbrack {i - \vartheta} \rbrack}s*\lbrack {i - \vartheta + N} \rbrack}}} )^{2}}} & (39)\end{matrix}$

For sufficiently large block lengths (N), the above term can be wellapproximated as $\begin{matrix}{{{{var}(ɛ)} \cong \frac{( {N + L} )\sigma_{s}^{2}\sigma_{w}^{2}}{( {2\pi\quad L\quad\sigma_{s}^{2}} )^{2}}} = \frac{( {N + L} )}{( {2\pi\quad L} )^{2}{SNR}}} & (40)\end{matrix}$

The resemblance between the estimators of equations (21) and (33)reveals a unified structure which characterizes the ML, MVU and momentestimators which can be classified into a single unified structure. Thisprovides a unique framework for analysis of the proposed estimators.Moreover, it allows an investigation of the effect of symbol timingerror in the estimation of carrier offset for each individual estimator.Comparing the MVU estimator given in (21) to the moment estimator in(33) reveals some similarities in the structure of the estimators.Clearly, both moment and MVU estimators use the same mapping function,namely the log function, to project the data statistics into theestimation domain. The only difference is in the form of statistics usedfor each scheme.

FIG. 2 is a block diagram depiction of the ML estimator disclosed in thecross-referenced, copending application and the MVU and momentestimators of the instant invention. An autocorrelation operation on theobservation vector x[k] is performed by blocks 30 and 34; as shown inFIG. 2, both estimators of the instant invention obtain the statistic bycorrelating the samples with the Nth delayed samples. This operation isperformed by using a moving average (MA) filter (shown as a block 36) inthe structure of estimators. However, the MVU and moment estimators usedifferent upper and lower bound for the MA filter. In the momentestimator, the averaging is performed over the first N+L samples of theobservation vector. This would remove the requirement of knowing theexact timing offset parameter in estimation of carrier frequency offset.However, the estimate obtained from using this estimator results in aless accurate estimate (more variance) in comparison to MVU estimate. Onthe other hand, the MVU estimator requires the knowledge of symboltiming in the estimation of carrier frequency offset, as shown in FIG. 2by a block 38 conducting

to the MA block 36. Finally, as in FIG. 1, a block 40 and a block 42perform the natural logarithm and imaginary, respectively, operations onthe symbol stream before generating the estimate of carrier frequencyoffset ε.

Although the resemblance between MVU and ML estimators may not be asevident as that of MVU and moment estimator, it can be shown that MLestimator can also be classified into the same family. Knowing the factthat ℑ{log T_(1(x,))}=<T₁(x,

) the ML estimator can be expressed as $\begin{matrix}{ɛ_{ML} = {\frac{- 1}{2\pi}{\mathfrak{J}}\{ {\sum\limits_{k = \alpha}^{\beta}{x*\lbrack k\rbrack{x\lbrack {k + N} \rbrack}}} \}}} & (41)\end{matrix}$where the parameters α and β are functions of

_(ML) and can be obtained from the expression for T₁(x,

), supra. Thus, the ML estimator falls into the same family ofestimators. FIG. 2 also displays in tabular form these estimators, theirrespective summation upper and lower bounds, and their respectiveaveraging intervals. It is noticed that the ML estimator provides theupper and lower bound of the moving average filter (36) by extractingthe timing parameter from the likelihood function. Although the MLestimator has the advantage of exploiting the entire bandwidth byremoving the requirement for having pilot tones, the symbol timingestimate obtained from the ML estimator has a larger confidenceinterval. This may result in a considerable performance degradation incomparison to the pilot-based schemes.

Computer simulation is used to assess the performance of the threeestimators for synchronization of an OFDM system. The variance ofestimator is used as a performance measure through the study. Thesimulation parameters used are typical of the OFDM and digital audiobroadcast (DAB) environments. More specifically, the chosen FFT size (N)for OFDM is 64. Unless otherwise specified, the length of cyclic prefix(L), signal to noise ratio, and frequency offset are set to 8, 20 dB and0.01, respectively. Monte Carlo simulation is used to evaluate theperformance of the three estimators.

FIG. 3 graphically depicts a comparison between the performance of theML frequency offset estimator disclosed in the related, copendingapplication with the prior art ML frequency offset estimator given inthe van de Beek, et al, reference over the range of timing offsetparameter (

ε[1, N+L]).

FIG. 4 graphically depicts the performance of the MVU frequency offsetestimator of the instant invention under complete knowledge of timingoffset error. A careful examination of the variances reveals that thegap between MVU estimator and CRLB tends to zero as SNR increases. Alsoas illustrated in FIG. 4, the departure from CRLB happens rapidly as SNRgoes below a threshold. The threshold also depends on the length of CPand is moved toward lower SNRs as L increases. This can be justified interms of having more observation samples in estimating the unknownparameter. The choice of cyclic prefix length L represents a tradeoffbetween data rate reduction and performance (lower variance). IncreasingL brings the performance of MVU estimator closer to the CRLB,nevertheless, it could result in a considerable data rate reduction dueto the redundancy introduced by CP.

FIG. 5 graphically depicts the closed loop performance of the MVUestimator of the instant invention by comparing the frequency offsetestimate of the closed loop MVU estimator and the analytical derivationgiven in equation (28). It is clear that the simulation result veryclosely resembles the analytical model, thus consolidates theapproximate model of the closed loop system.

Numerous variations and modifications will become evident to thoseskilled in the art once the disclosure is fully appreciated. It isintended that the following claims be interpreted to embrace all suchvariations and modifications.

1-12. (canceled)
 13. A method of estimating carrier frequency offseterror in a received sample bit stream including an observation vector(OV), having an observed carrier frequency timing offset ε, and aplurality of data-symbol frames, said OV comprising an L-bit cyclicextension portion and a first and a second N-bit synchronization frame,comprising the steps of: selecting one estimator method from a group ofthree: a maximum likelihood (ML) estimator, a minimum variance unbiased(MVU) estimator, a moment estimator, based on said OV; and generatingfrom said selected estimator an estimate of carrier frequency offseterror.
 14. The carrier frequency offset error estimation method of claim13 wherein said MVU estimator method comprises the steps: generating aprobability density function (PDF) based on said OV; and generating fromsaid PDF an estimate of carrier frequency offset error, ε_(MVU|), beingsaid MVU estimator.
 15. The carrier frequency offset error estimationmethod of claim 14 wherein said OV comprises an L-bit cyclic extensionportion and a first and a second N-bit synchronization frame, andwherein said PDF comprises a first term, p1, based on said timing offset

being within the span 1 to N and a second term, p2, based on said timingoffset

being within the span N+1 to N+L.
 16. The carrier frequency offset errorestimation method of claim 15 wherein said received bit stream hasuncorrelated independent identically distributed random signal and noisesequence variables with power of σ_(s) ² and σ_(n) ², respectively,wherein said OV is denoted x, and wherein said MVU estimator ε_(MVU|) isthe conditional expectation of a second moment estimator, said secondmoment estimator given by$\overset{\Cup}{ɛ} = {\frac{??}{2\pi}\ln{\{ {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \upsilon}^{\vartheta + L - 1}{{x\lbrack k\rbrack}x*\lbrack {K + N} \rbrack}}} \}.}}$17. The carrier frequency offset error estimation method of claim 16wherein said MVU estimator ε_(MVU|) is given by $\begin{matrix}{{\overset{\Cup}{ɛ}}_{{MVU}|\vartheta} = {E(  \overset{\Cup}{ɛ} |_{T_{1{({\chi,\vartheta})}}} )}} \\{= {\frac{??}{2\pi}\ln\quad E\{  {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}{{x\lbrack k\rbrack}x*\lbrack {k + N} \rbrack}}} |_{T_{1}{({\chi,\vartheta})}} \}}} \\{= {\frac{1}{2\pi}{??}\{ {\ln\frac{T_{1}( {\chi,\vartheta} )}{L\quad\sigma_{s}^{2}}} \}}}\end{matrix}$ where ℑ is the imaginary operator and where$T_{{1{({\chi,\vartheta})}})} = \{ \begin{matrix}{\sum\limits_{k = \vartheta}^{L + \vartheta - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} & {1 \leq \vartheta \leq N} \\{{\sum\limits_{k = 0}^{\vartheta - N - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} + {\sum\limits_{k = \vartheta}^{N + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} & {{N + 1} \leq \vartheta \leq {N + L}}\end{matrix} $
 18. The carrier frequency offset estimation methodof claim 13 wherein said moment estimator method comprises the steps:generating the expected value of the autocorrelation of the kth entry ofsaid OV; and generating from said expected value an estimate of carrierfrequency offset error, ε_(mom), being said moment estimator.
 19. Thecarrier frequency offset error estimation method of claim 18 whereinsaid OV is denoted x, and wherein said moment estimator ε_(mom) is givenby${\hat{ɛ}}_{mom} = {\frac{??}{2\pi}\{ {\ln\quad{T_{3}(\chi)}} \}}$where ℑ is the imaginary operator, and where the statistic T₃(χ) isgiven by${T_{3}(\chi)} = {\frac{1}{L\quad\sigma\frac{2}{s}}{\sum\limits_{k = 0}^{N + L + 1}\quad{{x\lbrack k\rbrack}{{x^{*}\lbrack {k + N} \rbrack}.}}}}$20. Apparatus for estimating carrier frequency offset error in areceived sample bit stream including an observation vector (OV), havingan observed carrier frequency timing offset ε, and a plurality ofdata-symbol frames, said OV comprising an L-bit cyclic extension portionand a first and a second N-bit synchronization frame, comprising: meansfor selecting one estimator method from a group of three: a maximumlikelihood (ML) estimator, a minimum variance unbiased (MVU) estimator,a moment estimator, based on said OV; and means for generating from saidselected estimator an estimate of carrier frequency offset error. 21.The invention of claim 20, wherein said MVU estimator method comprises:generating a probability density function (PDF) based on said OV; andgenerating from said PDF an estimate of carrier frequency offset error,ε_(MVU|), being said MVU estimator.
 22. The invention of claim 21,wherein said OV comprises an L-bit cyclic extension portion and a firstand a second N-bit synchronization frame, and wherein said PDF comprisesa first term, p1, based on said timing offset

being within the span 1 to N and a second term, p2, based on said timingoffset

being within the span N+1 to N+L.
 23. The invention of claim 22, whereinsaid received bit stream has uncorrelated independent identicallydistributed random signal and noise sequence variables with power ofσ_(s) ² and σ_(n) ², respectively, wherein said OV is denoted x, andwherein said MVU estimator ε_(MVU|) is the conditional expectation of asecond moment estimator, said second moment estimator given by$\overset{\Cup}{ɛ} = {\frac{??}{2\pi}\ln\quad{\{ {\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \upsilon}^{\vartheta + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} \}.}}$24. The invention of claim 23, wherein said MVU estimator ε_(MVU|) isgiven by $\begin{matrix}{{{\overset{\Cup}{ɛ}{MVU}}❘\vartheta} = {E( {\overset{\Cup}{ɛ}❘_{T_{1}{({\chi,\vartheta})}}} )}} \\{= {\frac{??}{2\pi}\ln\quad E\{ {{\frac{1}{L\quad\sigma_{s}^{2}}{\sum\limits_{k = \vartheta}^{\vartheta + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}}❘_{T_{1}{({\chi,\vartheta})}}} \}}} \\{= {\frac{1}{2\pi}{??}\{ {\ln\frac{T_{1}( {\chi,\vartheta} )}{L\quad\sigma_{s}^{2}}} \}}}\end{matrix}$ where ℑ is the imaginary operator and where$T_{{1{({\chi,\vartheta})}})} = \{ \begin{matrix}{\sum\limits_{k = \vartheta}^{L + \vartheta - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} & {1 \leq \vartheta \leq N} \\{{\sum\limits_{k = 0}^{\vartheta - N - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}} + {\sum\limits_{k = \vartheta}^{N + L - 1}\quad{{x\lbrack k\rbrack}{x^{*}\lbrack {k + N} \rbrack}}}} & {{N + 1} \leq \vartheta \leq {N + L}}\end{matrix} $
 25. The invention of claim 20, wherein said momentestimator method comprises the steps: generating the expected value ofthe autocorrelation of the kth entry of said OV; and generating formsaid expected value an estimate of carrier frequency offset error,ε_(mom), being said moment estimator.
 26. The invention of claim 25,wherein said OV is denoted x, and wherein said moment estimator ε_(mom)is given by${\hat{ɛ}}_{mom} = {\frac{??}{2\pi}\{ {\ln\quad{T_{3}(\chi)}} \}}$where ℑ is the imaginary operator, and where the statistic T₃(χ) isgiven by${T_{3}(\chi)} = {\frac{1}{L\quad\sigma\frac{2}{s}}{\sum\limits_{k = 0}^{N + L + 1}\quad{{x\lbrack k\rbrack}{{x^{*}\lbrack {k + N} \rbrack}.}}}}$